113 
where we suppose the co-ordinates rectangular; for if they 
were oblique, by transforming them to rectangular co-ordinates 
we should obtain an equation of the same degree as the 
above (Art. 96), and which could not therefore be more 
general than the one which we have assumed. We shall 
prove, as affirmed at Art. 53, that this equation, after being 
simplified as much as possible, will always assume one or 
other of the forms 
A.x 2 + By 8 + Cx 2 = D, 
By 2 + Cx 2 — 9, A' cc, 
the co-ordinates being rectangular ; and therefore the general 
equation of the second degree can never represent any other 
surface than one of those discussed in Section 2. 
140. Every surface of the second order has at least 
one diametral plane which is perpendicular to the chords 
bisected by it. 
Let the equation to the surface be 
atV 2 + by 2 +cx 2 +2a'yx+2b'x i v+2c'cvy+2a"a;+2b"y+2c"x + d = 0, 
and ,v = mx, y = nx, the equations to the line to which a 
system of chords is parallel; then the equation to the plane 
which bisects the chords is (Art. 137) 
{am + cn + b') jo + {bn + c'm + a) y + {c + b'm + an)x 
+ a" m + b" n + c" = 0, 
and our object is to shew that real values can be assigned 
to m and n, such that this plane shall be perpendicular to 
the chords. The conditions for this are (Art. 25) 
am -f c n + b' bn + cm + a! 
m = } —, n— 
c + bm + an c + 6m+ an 
from which, by eliminating one of the unknown quantities 
m or n, we shall obtain a cubic equation which will always 
give a real Value for the other; and the direction of the 
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